118 research outputs found
A uniform bound on the nilpotency degree of certain subalgebras of Kac-Moody algebras
Let be a Kac-Moody algebra and be Borel subalgebras of opposite signs. The intersection
is a finite-dimensional
solvable subalgebra of . We show that the nilpotency degree of
is bounded from above by a constant depending
only on . This confirms a conjecture of Y. Billig and A. Pianzola
\cite{BilligPia95}
Automorphism groups of right-angled buildings: simplicity and local splittings
We show that the group of type-preserving automorphisms of any irreducible
semi-regular thick right-angled building is abstractly simple. When the
building is locally finite, this gives a large family of compactly generated
(abstractly) simple locally compact groups. Specializing to appropriate cases,
we obtain examples of such simple groups that are locally indecomposable, but
have locally normal subgroups decomposing non-trivially as direct products.Comment: 26 pages. Several points were clarified and a few lemmas were added,
in accordance with the referee's repor
A sixteen-relator presentation of an infinite hyperbolic Kazhdan group
We provide an explicit presentation of an infinite hyperbolic Kazhdan group
with generators and relators of length at most . That group acts
properly and cocompactly on a hyperbolic triangle building of type .
We also point out a variation of the construction that yields examples of
lattices in -buildings admitting non-Desarguesian residues of
arbitrary prime power order.Comment: 9 pages, 1 figur
Amenable groups and Hadamard spaces with a totally disconnected isometry group
Let be a locally compact Hadamard space and be a totally disconnected
group acting continuously, properly and cocompactly on . We show that a
closed subgroup of is amenable if and only if it is (topologically locally
finite)-by-(virtually abelian). We are led to consider a set \bdfine X which
is a refinement of the visual boundary \bd X. For each x \in \bdfine X, the
stabilizer is amenable.Comment: 15 page
Rank one isometries of buildings and quasi-morphisms of Kac-Moody groups
Given an irreducible non-spherical non-affine (possibly non-proper) building
, we give sufficient conditions for a group G < \Aut(X) to admit an
infinite-dimensional space of non-trivial quasi-morphisms. The result applies
to all irreducible (non-spherical and non-affine) Kac-Moody groups over
integral domains. In particular, we obtain finitely presented simple groups of
infinite commutator width, thereby answering a question of Valerii G. Bardakov
from the Kourovka notebook. Independently of these considerations, we also
include a discussion of rank one isometries of proper CAT(0) spaces from a
rigidity viewpoint. In an appendix, we show that any homogeneous quasi-morphism
of a locally compact group with integer values is continuous.Comment: 19 pages; some typos have been corrected and the list of references
update
Indicability, residual finiteness, and simple subquotients of groups acting on trees
We establish three independent results on groups acting on trees. The first
implies that a compactly generated locally compact group which acts
continuously on a locally finite tree with nilpotent local action and no global
fixed point is virtually indicable; that is to say, it has a finite index
subgroup which surjects onto . The second ensures that irreducible
cocompact lattices in a product of non-discrete locally compact groups such
that one of the factors acts vertex-transitively on a tree with a nilpotent
local action cannot be residually finite. This is derived from a general
result, of independent interest, on irreducible lattices in product groups. The
third implies that every non-discrete Burger-Mozes universal group of
automorphisms of a tree with an arbitrary prescribed local action admits a
compactly generated closed subgroup with a non-discrete simple quotient. As
applications, we answer a question of D. Wise by proving the non-residual
finiteness of a certain lattice in a product of two regular trees, and we
obtain a negative answer to a question of C. Reid, concerning the structure
theory of locally compact groups
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